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# Exam 2 Study Guide & solutions MATH 2211

GSU

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This 8 page Study Guide was uploaded by hafsah qazi on Monday October 5, 2015. The Study Guide belongs to MATH 2211 at Georgia State University taught by van Der holst in Summer 2015. Since its upload, it has received 28 views. For similar materials see Calculus of One Variable l in Mathematics (M) at Georgia State University.

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Date Created: 10/05/15

Math 2211 Spring 2009 Test 2 Review Differentiation formulas Let x and gx are differentiable functions Then Lagrange prime notation f 1 gym fx i 8x39 fix 1 90 a f 39x a f x a f x a is a real number f 39 g x f x 39 gx39 f39xgx g39xfx I f xgx g2xfx gx71 0 g gx gx Leibniz notation d d d EMU i gx TU 96 t Eig E x 6 1 df afx afx 05 a is a real number dx i 39 3 i 22g dxfx gx dxfx gxdxgx fx qudx A d d df dg i for gfx garage fut k 2 2 agx7 0 1 gm gen gx Derivatives of higher order d2 1 i it fx A dryer d3 d d2 Ewe EFfx d fxii dbl n foo rzisa ositiveinte er dx a chmquot 39 p g g The derivative as a rate of change If y fx is a differentiable function then the derivative Cb it f 0 gives the rate of change of y with respect to x and this rate of change varies from point to point 39 The chain rule If y is a differentiable function ofu and u in turn is a differentiable function of x then y is a composite function of x and and so on Theorem 356 If g is differentiable at x and f is differentiable at gx then the composition f o g is differentiable at x and f 08 fgx39 f 398xg 0 Table of derivatives du xn nxn l a nun l ab du smx cosx smu cos u e Cbc u cosx smx cosu smu dx 2 2 dt tan x sec x tan u sec u 2 2 du cot x csc x cot u csc u C du sec x sec x tan x sec u sec at tan u dx du csc x csc x cot x csc u csc u cot u Ex Theorem 411 The meanvalue theorem If f is differentiable on the open interval a b and continuous on the closed interval a b then there is at least one number 0 in a b for which i b 39 7 fcf fa b 1 Theorem 412 Suppose that f isdifferentiable at x0 If f x0 gt 0then fx0 hltfx0 ltfxo 11 for all positive 11 suf ciently small If f 39xo lt 0then fxo h gt x gt fx0 h for all positive h suf ciently small Theorem 413 Rolle s theorem Suppose that f is differentiable on the open interval a b and continuous on the closed interval a b If a and fb are both 0 then there is at least one number c in a b for which f39c 0 9 H 53 6amp7quot quot l E a a a 391 r it s 39U as a 0 J 1 a r E J as V a Find the derivative of the given function Here you do not need to simplify the answer 1cosa 3 ft b 910 it2 sinl4m c Mar 2 4x9 c0ssin Find the indicated derivative of the given function Simplify your answer as much as possible 36 5x4221 a for h w 3 f4a 2 b gt 4sec5t g 7r2 N ONG7 MGM 2 a I u 2 39 C y 602 1 sin2 3 6x 3 w 5 if i fquot 7195 Ci 55 quot i am e Let f 24 213 36332 Find all values of a for which a f m 0 b f 8 gt 0 c f c lt 0 d Use implicit differentiation to obtain y in terms of a and y for 1 2 EC 5391 d5 v quot fast x2 2ymy2y21 Find an equation of the tangent line to the graph of 4112 632 10 at the point 1 1 Verify that f as 3E 4w satis es the conditions of the Mean Value Theorem for the interval 1 4 Find all numbers c that satisfy the conclusion of the theorem Show that f 39 14 2a2 8 satis es the conditions of Rolle s theorem on the interval 2 2 Find all numbers c on the interval for which f c 0 Suppose that f is differentiable on 2 6 and continuous on 26 Given that 0 g fquotL39 g 2 for all a in 2 6 Show that o 3 n6 W s 8 1 ats of Change 2L3 7 L f f x 5 Sxak Jr K5 Ty 739 g39i39i In 77 r niaf5 g g i LH 2k IS 7 x2 Rd N f H 2 f U H 9 up 11117L g 5 r X f C b LE 1 7 0 L 7 7 4173777 7 I ff c9733 7 gfgfnx 1 ME Q3 gig at 1 SM L 1 mos 5 l11 Tr 4 L J I D I k XS 7 V V quot quot 39 Smx 1W E ceg w 315 amp I m Z L r ufstm x 3m x cmr Ilj Sim X a god x Z Hmmg g39 l0 x2 Wm 1 a 2 WW w KM 4X COS SM g gt A J Leann i 1 3 CH i H r g QWCS UC ATHKL lCX B g r 7 Q39Cxi39 m wi cosca ee e m x f x 17 3 er w 1 1 K h I w 3 WU 3U LQS l gen f4rJCJ gi39smxsm Hi 05 x 337 J 5715 g i f w 5 65 4 32 d I 7 L39UJ L 31 397 f 5 f i J 39 Tj ckij f 331 14 WWE UL 7QCBS A 39L E7 31 7w1 39 7 t f f S 39 5 K 051 j S 13 a i P K t V 5161 CPS 39 Sic CK 7 i k 94 39 C Sa w 5quot f z f d i if 4px Sfquot Cr f f 4 f a 7777 y HT MW 7 m i 3 t 2 13 quotCOW 33913 57955 W v M t 7 y r 7 Arr 1K r WAJQJ 7 3 I g FL 7 f r 39 1 r 20 QSECS quot J39Sf Uf39ZO Cam St v Tam S I EEC LSAT 5 7 7 ft I i qr Z r iv f m quota F 339 39 V quotquot w 3 J 1 J 0 ms 2 P i 7 J Z 7 1 2 I I x n 1 151 8 X a 1 J 26 quotfig quotEw n 1 r 39 smzug m w39ruix39v air513117 7 W aw ma Ewbdngquot i gm 25 b cm I aim 2M mgA 7 m mfx e wij W owata39m WW Chi 39 01X k j mm S quot W r P aquot 13in A 39m if 55 g PINE 5 w z 1 m j L x J f 399 i iiii i 7 i LU w IInlu r rlfux m 2 39339 it f Lr ksv7 kfj gtLL L r d 31 f 1 oi WgL i 11 le 3 4 Hm10 3 Sir X 3 f m M 51 GT amp VLJ J D1 L quot 41 44 f I d 2 quot 2 ms 10 W O 7 7 N h 9 g GA 1 397quot 7 l a v a i w oh fem b5 swim Y quot a Haw1 b0 50 2v i 3V LJHCMCf M iii 9 5 cC2lt 4r 1 mm H S gibcl 70 H WTW L E mwm L j a awn WW1 a h f 3X36Qx971ampw igwb dn k f mwz rm f H g lube O7X1Dibltrjl a W ixfxfg 39er33 90 L43 MPua m ii 0Y5 X L tquot him 39 xikka W k 7 1 7 392 3 3 i f I x a 7 W PDQ if N 7 W 7 PM 0 20 i 1 3i i LAltic 13 U 1 QTS 40 7 7 me kiwi M HZ H r i 777V 7 7 n 0 mom she m zva KM gt m 0 wr v m 5 a W wigw W 39 739quot 3quot quot 11W W 39 cw 4 i W gt 1157 2K3 quot X ZKNL3g T3 3 w i 343 LA 2 at J J F 5 3 i J W 3t m u 739 i 75501 J J Z iv w 739 m a J 7 f My 22K 3K 5Uquot 2 quot hi i W it Q g Villa1er 7 7 jg f Hz 40147quot 3 EU J M 04gt 133 EX 7 l7u v da Q r Om m L f mmm mw kTagb M g fi39 rd g kyiiwx ranr E v1 E 414 a 145 2 307 f IS O W memf zuz4 n UT L a mi quotHm LLOLLS an Jym iv A1 m4me m 175 I 37 Conf mmnm 3 an I l 917 quot 7 r x 4139 A 1 friiiiaiipw a 3341 4 may x Mg C lFlVgSi f iquot 393 K 391 E Jr 3 3 k vr a LLJC H i r L

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